Iber v3. Reference manual and GUI. Non-Newtonian shallow flows calculation module


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==Iber v3. Reference manual and GUI. Non-Newtonian shallow flows calculation module==

Marcos Sanz-Ramos<sup>1</sup>, Ernest Bladé<sup>1</sup>

<sup>1</sup>Flumen Institute, Universitat Politècnica de Catalunya (UPC - Barcelona TECH) – International Centre of Numerical Methods in Engineering (CIMNE), 08034 Barcelona, Spain

''Correspondence to'': [mailto:marcos.sanz-ramos@upc.edu marcos.sanz-ramos@upc.edu]

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==Abstract==

Iber is a two-dimensional hydraulic model for the simulation of free surface flow in rivers and estuaries, and the simulation of environmental processes in fluvial hydraulics. Since the release of the first version of Iber, which included a hydrodynamic calculation engine fully coupled with sediment transport processes and turbulence, it has evolved to become a free surface flow modelling tool for highly complex environmental processes. This document presents the developments made for version 3, specifically, for the new calculation module for the simulation of shallow non-Newtonian flows called IberNNF. The shallow water equations are solved using an ad-hoc numerical scheme focused on the simulation of this type of flow in nature (e.g., steep slopes, irregular geometries). The graphical user interface (GUI) has been adapted to the module's new features to achieve a simple and user-friendly workflow.

'''Keywords''': non-Newtonian flows, snow avalanches, mud/debris flows, hypercongested flows, Iber

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==Resumen==

Iber es un modelo numérico bidimensional de simulación de flujo turbulento en lámina libre en régimen no-permanente, y de procesos medioambientales en hidráulica fluvial. Desde el lanzamiento de la primera versión, que incluía un motor de cálculo hidrodinámico para completamente acoplado con procesos de transporte de sedimento y turbulencia, Iber ha ido evolucionando hasta convertirse en una herramienta de modelización de flujo de agua en lámina libre de procesos ambientales de elevada complejidad. En este documento se presentan los desarrollos realizados para la versión 3, concretamente para el nuevo módulo de cálculo para la simulación de flujos no-newtonianos someros denominado IberNNF. La resolución de las ecuaciones de aguas someras se lleva a cabo mediante un esquema numérico propio enfocado a la simulación de este tipo de flujos en la naturaleza (p.ej., pendientes elevadas, geometrías irregulares). La interfaz gráfica de usuario (GUI) se ha adaptado para con las nuevas características del módulo con el fin de obtener un flujo de trabajo sencillo y amigable.

'''Palabras clave''': flujos no-newtonianos, aludes, flujo de lodos/escombros, flujos hipercongestionados, Iber

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==1 Introduction==

Numerical modelling of natural phenomena, particularly weather-related which are the 90 % of global disasters, is essential to analyse and predict hazardous situations for the people, the economy and the environment. The evolution of these numerical tools, from simple one-dimensional to complex three-dimensional models, to simulate hydrological hazards like floods, mass movements, and avalanches is challenging, especially those in which the fluid can be characterized as non–Newtonian flows.

IberNNF was developed, based on Iber [1], as a depth-averaged two-dimensional hydrodynamic numerical tool to simulate non–Newtonian shallow flows. To that end, a particular numerical scheme based on an upwind discretisation to ensure a proper balance between the non–velocity-dependent terms of the shear stresses and the pressure forces has been developed [2]. This ensures the stop of the fluid according to the rheological properties of the fluid, even in steep slopes and complex geometries. The code besides being validated and applied in theoretical, analytical, and real situations of common and non–common non–Newtonian shallow flows [2–7], it has been fully integrated in the graphical user interface of Iber. This facilitates the model build-up, setup and results visualization converting the new code in a software suite fully operational for all practitioners.

The common applications of IberNNF are the numerical modelling of dense snow avalanches, mine tailings propagation (e.g., after a dam-break), lahars, and hypercongested flows (e.g., wood laden flows). To that end, several rheological models were implemented making IberNNF versatile and widen applicable for non-Newtonian shallow flows.

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==2 Graphical user interface of IberNNF==

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===2.1 Generalities===

The current version of IberNNF is fully integrated into Iber. Thus, the same properties, options and main workflow used in Iber also applies to IberNNF. Only particular characteristics of this module are described below. Further information can be found in the Iber v3 Refence manual [8].

It is worth noticing that IberNNF currently works as an independent hydrodynamic module. None interaction between the rest of calculation modules is permitted due to the kind of flow that IberNNF simulate is not water. Future interactions are not discarded.

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===2.2 Particularities===

IberNNF, as for the rest of modules, must be activated. The activation of IberNNF can be done by:

:* The menu Iber tools >> Plug-ins…

:* The shortcut  [[Image:Draft_Sanz-Ramos_617790713-image1.png|12px]] (located on the left side of the interface, by default)

Once selected ‘NonNewtonian fluid’ as a module, and then applied, the interface will be adapted to this new hydrodynamic module oriented to simulate non–Newtonian flows. The main difference between Iber and IberNNF module relays on the implementation of the flow resistant terms, or rheological model, which have been split in two:

:* Velocity-dependent, available through Data >> Roughness >> Friction slope…

:* Non–Velocity-dependent, available through Data >> Problem data > Non-Newtonian Fluid

This separation is a consequence of the numerical scheme developed ad hoc for IberNNF. More information is available in Sanz-Ramos et al., 2023 [2].

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===2.3 Implementation of rheological properties of the fluid===

As mentioned previously, there is a different way to implement the rheological properties of the fluid in IberNNF.

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====2.3.1 Velocity-dependent terms====

Velocity-dependent terms of the rheological model must be implemented as a friction slope at each mesh element (Data >> Roughness >> Friction slope…). These parameters can be defined manually or automatically (by a raster file), and are associated to the concept known as ‘Land use’; thus, they can vary spatially.

{| style="width: 84%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
|-
|  style="text-align: center;width: 50%;"|[[Image:Draft_Sanz-Ramos_617790713-image2.png|204px]] 
|  style="text-align: center;width: 50%;"|[[Image:Draft_Sanz-Ramos_617790713-image3.png|204px]] 
|-
|  style="text-align: center;"|(a)
|  style="text-align: center;"|(b)
|}


<span style="text-align: center; font-size: 75%;">'''Fig. 1. Land uses windows: (a) database of land uses for non-Newtonian flows; (b) list of velocity-dependent parameters according to each rheological model.'''</span>

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====2.3.2 Non–Velocity-dependent terms====

By contrast, non–Velocity-dependent terms can be interpreted as a characteristic of the fluid; thus, they cannot vary spatially –perhaps temporally– and they must be defined as a constant value (Data >> Problem data > Non Newtonian Fluid). This is the case of the flow density, the pressure factor, the Coulomb friction coefficient, the yield stress, etc.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 [[Image:Draft_Sanz-Ramos_617790713-image4.png|306px]] </div>

<span id='_Hlk202869113'></span><span style="text-align: center; font-size: 75%;">'''Fig. 2. Problem data window. Non-Newtonian fluid tab allows the selection of the rheological model to be used and other properties.'''</span>

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===2.4 Stop criterion===

The detention of any fluid is consequence of a balance between resistance and driving forces. IberNNF uses an ad hoc numerical scheme that allows the stop of the fluid according to the fluid properties [2], i.e. the rheological model.

Another popular numerical model uses a stopping criterion based on controlling the momentum, where the fluid is made to stop when its momentum is lower than a user-defined fraction of its maximum momentum. However, this criterion lacks a physical basis, as the maximum momentum depends on the avalanche’s characteristics at very different location and time than those when it stops.

Both stop criterion are implemented into IberNNF; nevertheless, '''we encourage to use the ‘Rheology based’ criterion because is physically based'''.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 [[Image:Draft_Sanz-Ramos_617790713-image5.png|306px|alt=''''Fig. 3. Problem data window. Selection of the stop criterion.''''|'''Fig. 3. Problem data window. Selection of the stop criterion.''']] </div>

<span style="text-align: center; font-size: 75%;">'''Fig. 3. Problem data window. Selection of the stop criterion.'''</span>

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==3 Governing equations==

This section is a brief description of the governing equations of IberNNF. Further details about this hydrodynamic module and the numerical scheme used to solve the equations can be found in Sanz-Ramos et al. 2023 [2].

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===3.1 2D shallow water equations for non-Newtonian shallow flows===

IberNNF solves a particular case of the two-dimensional shallow water equations (2D-SWE), a hyperbolic nonlinear system of three partial differential equations described in Equation <span id='cite-_Ref202869890'></span>[[#_Ref202869890|(1)]]:

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>\begin{matrix}\frac{\partial h}{\partial t}+\frac{\partial {q}_{x}}{\partial x}+\frac{\partial {q}_{y}}{\partial y}=E\\\frac{\partial {q}_{x}}{\partial t}+\frac{\partial }{\partial x}\left( \frac{{q}_{x}^{2}}{h}+g'\frac{{h}^{2}}{2}{K}_{p}\right) +\frac{\partial }{\partial y}\left( \frac{{q}_{x}{q}_{y}}{h}\right) =g'h\left( {S}_{o,y}-{S}_{f,x}\right) \\\frac{\partial {q}_{x}}{\partial t}+\frac{\partial }{\partial x}\left( \frac{{q}_{x}{q}_{y}}{h}\right) +\frac{\partial }{\partial y}\left( \frac{{q}_{y}^{2}}{h}+g'\frac{{h}^{2}}{2}{K}_{p}\right) =g'h\left( {S}_{o,y}-{S}_{f,y}\right) \end{matrix}\,</math> 
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref202869890'></span>(1)
|}


<span id='_Hlk123797141'></span>where <math display="inline">h</math> is the water depth, <math display="inline">{q}_{x}</math> and <math display="inline">{q}_{y}</math> are the two components of the specific discharge, <math display="inline">g</math> is the gravitational acceleration, <math display="inline">{S}_{o,x}</math> and <math display="inline">{S}_{o,y}</math> are the two bottom slope components computed as <math display="inline">{\mathit{\boldsymbol{S}}}_{\mathit{\boldsymbol{o}}}=</math><math>{\left( \frac{\partial {z}_{b}}{\partial x},\frac{\partial {z}_{b}}{\partial y}\right) }^{T}</math>, where <math display="inline">{z}_{b}</math> is the bed elevation, and <math display="inline">{S}_{f,x}</math> and <math display="inline">{S}_{f,y}</math> are the two friction slope components computed throughout the rheological model. The friction forces exerted over an inclined bed and the pressure terms can be corrected by replacing the gravity acceleration <math display="inline">g</math> by <math display="inline">{g}^{'}=</math><math>\mathrm{g{cos}^{2}}\,\theta</math>  [9–11]. Since the hydrostatic and isotropic pressure distribution cannot be assumed for non-Newtonian flows, as it is done for free surface water flows [12], a factor <math display="inline">{K}_{p}</math> multiplying the pressure terms in the momentum equations was applied [13]. A <math display="inline">{K}_{p}</math> value equal to 1 implies hydrostatic and isotropic pressure distribution. The term <math display="inline">E</math> is entrainment, a process by which solid particles or fragments become incorporated into a moving fluid. The current code partially integrates entrainment formulas based on flow velocity criterion [14], flow height criterion [15] and bed shear stress criterion [16]. The acknowledgment of entrainment is essential for ensuring reliable outcomes and, thus, preventing the underestimation of the volume of snow descending a slope.

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===3.2 Rheological models===

Rheological models to describe both dynamic and static phase of non–Newtonian shallow flows exist for a wide field of applications. In particular, for those related to environmental flows, and more specially for shallow flows, several rheological models have been developed to describe the relationship between the shear stress and the shear rate [17].

From the simplest Potential law to the full –and complex– Bingham model, several rheological models exist in the literature, the development of each one being oriented to achieve a particular reproduction of a fluid behaviour. The aim of IberNNF is not to include as rheological models as possible –or exist–; however, there are some models that, although they have been omitted, can be easily integrated into the proposed numerical scheme by slightly adapting the code. This would allow a broader simulation of the behaviour of non–Newtonian shallow fluids.

Two hypotheses are usually considered in non-Newtonian shallow flows modelling: ''a monophasic fluid'', in which the fluid is formed by a unique phase where all components are perfectly mixed, and ''shear stress grouping'', in which the effect of different shear stresses are grouped as five components of a single term [18] as follows:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>\tau ={\tau }_{d}+{\tau }_{t}+{\tau }_{v}+{\tau }_{mc}+{\tau }_{c}</math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref202871261'></span>(2)
|}


where <math display="inline">{\tau }_{d}</math> represents the dispersive term, <math display="inline">{\tau }_{t}</math> the turbulent term, <math display="inline">{\tau }_{v}</math> the viscous term, <math display="inline">{\tau }_{mc}</math> the Mohr–Coulomb terms, and <math display="inline">{\tau }_{c}</math> the cohesive term. In these components, the appropriate rheological model for the particular purpose of each work is obtained by selecting one or several components of Equation <span id='cite-_Ref202871261'></span>[[#_Ref202871261|(2)]].

IberNNF integrates several rheological models to represent the resistance forces that act against flow motion of non–Newtonian flows, such as mudflows, debris flows, snow avalanches, lahars, etc. [2–7]. The following sections describe the rheological models implemented expressed in friction slope form ( <math display="inline">\tau =</math><math>\, \rho gh{S}_{f}</math>).

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====3.2.1 Manning====

<span id='_Hlk164577115'></span>The Manning rheological model, an empirical equation widely utilised in hydraulics and hydrology, applies to uniform flow in open channels and is a function of the channel velocity, flow area and channel slope:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>{S}_{f}=\frac{{n}^{2}{v}^{2}}{{h}^{\frac{4}{3}}}</math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(3)
|}


<span id='_Hlk164577165'></span>where <math display="inline">n</math> is the Manning coefficient, <math display="inline">v</math> is the flow velocity and <math display="inline">h</math> is the flow depth. It is related to turbulent friction ( <math display="inline">{\tau }_{t}</math>), being utilised by several authors for simulating hyperconcentrated flows [19–22]. The unique value for calibration is the Manning coefficient ( <math display="inline">n</math>).

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====3.2.2 Bingham (simplified)====

Since the proposal of the Bingham (1916) rheological model, several approaches have been introduced to deal with the difficulties on directly obtaining the shear stress proportional to the flow velocity [24]. Assuming an incompressible and homogeneous flow [25,26], the following expression for the viscous ( <math display="inline">{\tau }_{v}</math>) and the Mohr–Coulomb ( <math display="inline">{\tau }_{mc}</math>) contributions:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>{S}_{f}=\frac{1}{\rho gh}\left( \frac{3}{2}{\tau }_{y}+3\frac{{\mu }_{B}v}{h}\right)</math> 
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(4)
|}


where <math display="inline">{\tau }_{y}</math> is the yield stress, <math display="inline">\rho</math>  is the fluid density, <math display="inline">h</math> is the flow depth, <math display="inline">{\mu }_{B}</math> is the fluid viscosity, <math display="inline">v</math> is the flow velocity, and <math display="inline">g</math> is the gravitational acceleration.

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====3.2.3 Voellmy====

Voellmy (1955) presented a rheological model that considers the turbulent ( <math display="inline">{\tau }_{t}</math>) and the Mohr–Coulomb ( <math display="inline">{\tau }_{mc}</math>) terms as follows:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>{S}_{f}=\mu +\frac{{v}^{2}}{\xi h}</math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(5)
|}


<span id='_Hlk164436656'></span>where <math display="inline">\xi</math>  is the turbulent friction coefficient, <math display="inline">\mu</math>  is the Coulomb friction coefficient, <math display="inline">h</math> is the flow depth and <math display="inline">v</math> is the flow velocity.

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====3.2.4 Bartelt====

Bartelt et al. (2015) developed a new resistance term related to the cohesion, a physical property of the fluid. This rheological model is commonly used together with the Voellmy model, and expresses as follows:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>{S}_{f}=\frac{1}{\rho gh}\left( {C}_{B}\, \left( 1-\mu \right) \left( 1-{e}^{-\frac{\rho gh}{{C}_{B}\, }}\right) \right)</math> 
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(6)
|}


where <math display="inline">\rho</math>  is the fluid density, <math display="inline">g</math> is the gravitational acceleration, <math display="inline">h</math> is the flow depth, <math display="inline">{C}_{B}\,</math>  is the cohesion, and <math display="inline">\mu</math>  is the Coulomb friction coefficient.

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====3.2.5 Dilatant====

<span id='_Hlk164577197'></span>Similarly to the Manning rheological models, and considering constant sediment concentration and uniform flow, Macedonio and Pareschi (1992) derived the following expression: <math display="inline">\tau =</math><math>{\tau }_{y}+{\mu }_{1}{\left( \frac{dv}{dz}\right) }^{\alpha }</math>, where <math display="inline">{\tau }_{y}</math> is the yield stress, <math display="inline">{\mu }_{1}</math> is a proportionality coefficient and <math display="inline">\alpha</math>  is the flow behaviour index.

When <math display="inline">\alpha</math>  = 2 a dilatant flow behaviour is expected:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>{S}_{f}=\frac{{n}^{2}{v}^{2}}{{h}^{3}}</math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(7)
|}


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====3.2.6 Viscous====

Macedonio and Pareschi (1992) also presented the application of the Manning equation to viscous flows by particularizing the parameter <math display="inline">\alpha</math>  = 1. This allows for the representation of viscous flows:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>{S}_{f}=\frac{{n}^{2}v}{{h}^{2}}</math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(8)
|}


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====3.2.7 O’Brien====

On the other hand, O’Brien and Julien (1988) derived an expression for the representation of the shear stress of mudflows, being a quadratic equation that integrates the Mohr–Coulomb term ( <math display="inline">{\tau }_{mc}</math>), the viscous term ( <math display="inline">{\tau }_{v}</math>) and the turbulent term ( <math display="inline">{\tau }_{t}</math>) as follows:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>{S}_{f}=\frac{{\tau }_{y}}{\rho gh}+\frac{K{\mu }_{B}v}{8\rho g{h}^{2}}+\frac{{n}^{2}{v}^{2}}{{h}^{\frac{4}{3}}}</math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(9)
|}


<span id='_Hlk164577448'></span>where <math display="inline">{\tau }_{y}</math> is the yield stress, <math display="inline">\rho</math>  is the fluid density, <math display="inline">g</math> is the gravitational acceleration, <math display="inline">h</math> is the flow depth, <math display="inline">K</math> is a resistance parameter, <math display="inline">{\mu }_{B}</math> is the flow viscosity, <math display="inline">v</math> is the flow velocity, and <math display="inline">n</math> is the Manning coefficient.

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====3.2.8 Herschel-Bulkley====

The formulation of Herschel and Bulkley (1926) is a generalization of various expressions in which, depending on the value of the coefficient <math display="inline">\alpha</math> , dilatant, viscous, plastic, etc. behaviours can be derived. This formula follows the following expression:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>{S}_{f}=\frac{1}{\rho gh}\left( {\tau }_{y}+k{\left( \frac{v}{h}\right) }^{\alpha }\right)</math> 
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(10)
|}


where <math display="inline">{\tau }_{y}</math> is the yield stress, <math display="inline">\rho</math>  is the fluid density, <math display="inline">g</math> is the gravitational acceleration, <math display="inline">h</math> is the flow depth, <math display="inline">k</math> is a consistency parameter, and <math display="inline">v</math> is the flow velocity.

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===3.3 Entrainment===

The entrainment is a relevant phenomenon in non-Newtonian flow dynamic modelling because the shear stress between the moving fluid and the terrain generally erode the bottom. This eroded material is then aggregated to the bulk, and might affect it properties (e.g., fluid density) and behaviour.

The effects of entrainment extend beyond altering mass and energy balances. Predicted velocities along the bulk path and the kinetic energy upon reaching the runout zone are also affected. These changes directly influence runout distances and have substantial implications for hazard and risk mapping. Particularly for snow avalanche modelling, entrainment leads to higher predicted flow heights and volumes of avalanches [15,32–35].

Accurate predictions are crucial for designing infrastructure, such as barriers or dams, as incorrect estimations may result in inadequate protection or increased costs. Therefore, precise consideration of entrainment is essential for determining runout distances and optimizing infrastructure design to mitigate hazards effectively.

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====3.3.1 Velocity model====

This is a simple model that considers mass entrainment as function of the flow velocity. In contrast with another popular model, IberNNF considers entrainment when the flow velocity is greater than a threshold ( <math display="inline">{u}_{crit}</math>).

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>E=\left\{ \begin{matrix}0\, \\{K}_{u}\left( u-{u}_{crit}\right) \end{matrix}\begin{matrix}\\\end{matrix}\begin{matrix}when\, u\leq {u}_{crit}\\when\, u>{u}_{crit}\end{matrix}\right.</math> 
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(11)
|}


where <math display="inline">{K}_{u}</math> is the entrainment rate, which commonly range from 5 to 40·10<sup>-5</sup>.

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====3.3.2 Height model====

In this model, the entrainment depends on the load of the underlying snow cover as long as its height reaches a fixed minimum value ( <math display="inline">{h}_{crit}</math>); otherwise, the entrainment will be considered inexistent [15]. This model also integrates an upper limit for the height based on the dry friction law to avoid the dry friction increasing limitless [36]:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>E=\left\{ \begin{matrix}0&&when\, h\leq {h}_{crit}\\{K}_{h}\left( h-{h}_{crit}\right) &&when\, {h}_{crit}<h<{h}_{lim}\\{K}_{h}{h}_{lim}&&when\, h\geq {h}_{lim}\end{matrix}\right.</math> 
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(12)
|}


where <math display="inline">{K}_{u}</math> is the entrainment rate, which commonly range from 1 to 8·10<sup>-3</sup> s<sup>-1</sup>, and <math display="inline">{h}_{lim}</math> being the maximum avalanche flux height at which yielding at the basal surface occurs:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>{h}_{lim}=\frac{{\tau }_{lim}}{\mu \rho g\mathrm{cos}\,\theta }</math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(13)
|}


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====3.3.3 Squared velocity model====

This equation is similar to the velocity model although the entrainment rate is considered to vary with the squared velocity of the avalanche [15]:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>E=\left\{ \begin{matrix}0\, \\{K}_{u}^{2}\left( {u}^{2}-{u}_{crit}^{2}\right) \end{matrix}\begin{matrix}\\\end{matrix}\begin{matrix}when\, {u}^{2}\leq {u}_{crit}^{2}\\when\, {u}^{2}>{u}_{crit}^{2}\end{matrix}\right.</math> 
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(14)
|}


where <math display="inline">{K}_{u}^{2}</math> is the entrainment rate, which commonly range from 4 to 32·10<sup>-6</sup>.

<span id='_Toc203976079'></span>

====3.3.4 Bed shear stress model====

Similar to how the sediment transport is computed, a new equation to calculate the entrainment as a function of the bed shear stress between the lower snow layer and the avalanche:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>E=\left\{ \begin{matrix}0\, \\{K}_{\tau }\left( \tau -{\tau }_{crit}\right) \end{matrix}\begin{matrix}\\\end{matrix}\begin{matrix}when\, \tau \leq {\tau }_{crit}\\when\, \tau >{\tau }_{crit}\end{matrix}\right.</math> 
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(15)
|}


where <math display="inline">{K}_{\tau }</math> is the entrainment rate, which a range from 1.5 to 12·10<sup>-6</sup> m·s<sup>-1</sup>·Pa<sup>-1</sup> is proposed [16].

<span id='_Toc203976080'></span>

==4 Results==

As in the hydrodynamic module for water flows, IberNNF also integrates flow depths, velocities, elevation, etc. However, particular results can be activated through Data >> Problem data >> NonNewtonian fluid tab, such as extra topographical information (terrain slope) and impact forces [37,38]. This results essentially applies for dense snow avalanche modelling, but they are not limited to.

Particularly for impact forces, IberNNF calculates the dynamic pressure (Equation <span id='cite-_Ref202886400'></span>[[#_Ref202886400|(16)]]), the peak dynamic pressure (Equation <span id='cite-_Ref202886402'></span>[[#_Ref202886402|(17)]]) and its maximus as follows:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>p=\rho {u}^{2}</math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref202886400'></span>(16)
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>{p}_{peak}=3\rho {u}^{2}</math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref202886402'></span>(17)
|}


where <math display="inline">\rho</math>  is the fluid density and <math display="inline">u</math> is the fluid velocity.

<br/>

<br/>

<span id='_Toc203976081'></span>

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